The easily recognized eyes of a realistic anime character
Anime are the animated TV shows and movies based on manga, the Japanese equivalent of American comics and graphic novels. Anime, like manga, can be categorized by their audience---for example, shonen are action comics for boys, and shoujo are romance comics for girls---or by a genre such as magical girls, where young girls are fantasy superheroines, such as "Sailor Moon." But many people confuse these with the drawing styles themselves, which may be used in almost any genre aimed at any audience.
Classic Anime
When the legendary Osamu Tezuka created his groundbreaking series "Astroboy" back in the 1960s, he also created the traditional anime style. Based on Disney cartoons, he developed the look that has become the stereotype for all Japanese comics---large heads with huge eyes, small noses and mouths, and sometimes rubbery features. There was a humorous aspect to this look that has been slowly toned down to increase its range of expression but not enough to keep outrageous exaggeration from being used. Perhaps the most successful examples of this style are the various "Dragonball" video series, where a variety of semi-realistic characters interact with purely imaginary characters... and yet no character seems out of place.
Realistic Anime
Mention anime to most people, and they will probably think of this style. Christopher Hart, an animator and teacher who has written many books on manga and anime, refers to the most popular version of this style as the "high-tech punk look." In many ways, it resembles American superhero comics---the characters are slightly taller than normal humans, but the features and proportions are otherwise realistic enough that you can forget you are watching a cartoon. This style is used in a wide variety of anime genres with more serious topics, like "Akira" and other fantasy, science fiction and dramatic series.
Children's Anime
While anime often seems to be totally aimed at older children, teens and adults, anime for the smallest viewers also exists. This cute style takes the traditional style to a simplistic extreme---characters often seem to be little more than blobs with tiny arms and legs, huge eyes, small mouths, and sometimes no nose at all. Extremely short human characters called "chibi" sometimes star in these cartoons; these characters are almost all head. (Chibi are sometimes used in traditional anime, but they are usually parodies of existing characters and show up only briefly for comic effect when the characters are acting outlandishly.) Animal characters are extremely simple shapes with limited movements. The children's series "Hamtaro," about a group of small pets who call themselves "Ham-hams," is a good example.
Computer Anime
The use of computers in anime has created a style of its own. While sometimes mimicking the look of traditional Japanese animation, computer animation has allowed artists to use 3D modeling to give classic anime a look similar to a computer game. In other cases it has allowed the automation of the drawing process, eliminating some of the jerkiness associated with the low frame rates used to save money in traditional animation. This technology is still in its infancy, but the styles created with it are truly stunning. It is frequently used in mecha anime, such as the "Gundam" series and the "Final Fantasy" movies.